# Hill four-body problem with oblate tertiary: an application to the Sun-Jupiter-Hektor-Skamandrios system

## TL;DR

This paper analyzes a restricted four-body problem involving an oblate tertiary, deriving Hill's approximation, equilibrium points, and stability, with application to the Sun-Jupiter-Hektor-Skamandrios system.

## Contribution

It introduces a Hill's approximation for a four-body system with an oblate tertiary, extending classical models to more realistic celestial configurations.

## Key findings

- Triangular central configurations are isosceles due to oblateness.
- Derived Hill's equations for the infinitesimal body near the oblate tertiary.
- Identified equilibrium points and analyzed their stability.

## Abstract

We consider a restricted four-body problem with a precise hierarchy between the bodies: two point-mass bigger bodies, a smaller one with oblate shape, and an infinitesimal body in the neighborhood of the oblate body. The three heavy bodies are assumed to move in a plane under their mutual gravity, and the fourth body moves under the gravitational influence of the three heavy bodies, but without affecting them. We start by finding the triangular central configurations of the three heavy bodies; since one body is oblate, the triangle is isosceles, rather than equilateral as in the point mass case. We assume that the three heavy bodies are in such a central configuration and we perform a Hill's approximation of the equations of motion describing the dynamics of the infinitesimal body in a neighborhood of the oblate body. Through the use of Hill's variables and a limiting procedure, this approximation amounts to sending the two other bodies to infinity. Finally, for the Hill approximation, we find the equilibrium points of the infinitesimal body and determine their stability. As a motivating example, we consider the dynamics of the moonlet Skamandrios of Jupiter's Trojan asteroid Hektor.